※ 引述《aesth (應該很帥吧)》之銘言：
: ※ 引述《ed78617 (雞爪)》之銘言：
: : [領域]古典力學 (題目相關領域)
: : [來源] Goldstein (課本習題、考古題、參考書...)
: : [題目]
: : Show that a rotation about any given axis can be obtained as the product of
: : two successive rotations, each through 180°
: : [瓶頸] (寫寫自己的想法，方便大家為你解答)
: : 請教一下要如何下手
: : 是利用球面三角形的餘弦律和正弦律嗎？
: : 先謝謝各位~
: 太久沒碰古力了 我用量子力學證明
: Take two 3-d unit vector n1 and n2.
: in the case of n1˙n2 = cos(γ) and n1×n2=n3*sin(γ),
: γ the angle between two vectors and n3 is the unit vector orth. to n1 and n2.
: A vector V after a rotation around n1 by π
: becomes exp(-iσ˙n1*π) V exp(iσ˙n1*π/2), where σ is the Pauli matrix
: By using exp{iσ˙n1*π}=cos(π/2)+i*sin(π/2)*(σ˙n1)=i*(σ˙n1)
: (-1)*exp{iσ˙n1*π}*exp{iσ˙n2*π} = (σ˙n1)(σ˙n2)
: =1 (n1˙n2) + i σ˙(n1×n2) = cos(γ) + i (σ˙n3) sin(γ)
: = exp{i(σ˙n3)*γ},
: which stands for a rotation about the n3-axis by 2*γ.
剛好我欠文章數，哈
The rotation transformation is defined by a 3x3 matrix R
V'=RV, V=(Vx,Vy,Vz) and same as V', or Vi' = sum_{j} R_{i,j}*Vj
Before we have shown the rotaion in the SU(2) representation by using
the operator u3=u(n3,γ)=exp{i(σ˙n3)*γ}
({n3,γ} can be replaced in terms of Eularian angles)
Here we will relate R and u, that is, R=R(u)
Define a 2x2 matrix, m=σ˙V, V=(Vx,Vy,Vz)
and a rotation operation u1=exp(-iσ˙n1*θ1)
The rotated matrix, e.q. to vector V, is
u1*m*(u1)^-1 ≒ m'=σ˙V', V'=(Vx',Vy',Vz')
We will find that V' is the rotated vector of V.
m'=Vx'*σx + Vy'*σy +Vz'*σz
= u1*σ*(u1)^-1 ˙V
= Vx {u1*σx*(u1)^-1} + Vy {u1*σy*(u1)^-1} +Vz {u1*σz*(u1)^-1}
(σx, σy, and σz are three 2x2 Pauli matrices)
You can prove that,
R(u1)_{i,j} = 0.5*trace{(σi)*u1*(σj)*u1^-1}, i,j=1,2,3
After we relate R in SO(3) and u in SU(2).
You can also prove that
R(u1)R(u2) = R(u1*u2) for any u1, u2 belong to SU(2)
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